We present a new construction approach for symmetric lifted B-spline wavelets on irregular polygonal control meshes defining two-manifold topolgies. Polygonal control meshes are recursively refined by stationary subdivion rules and converge to piecewise polynomaial limit surfaces. At every subdivision level, our wavelet transforms provide an efficient way to add geometric details that are expanded from wavelet coefficients. Both wavelet decomposition and reconstruction operations are based on local lifting steps and have linear-time complexity.

We present a highly efficient, automtic method for the generation of hierarchical surface triangulations. Given a set of scattered points in three-dimensional space, without connectivity information, our method reconstructs a valid triangulated surface model in a two-step procedure. First, we apply clustering to the set of given points and identify point subsets in locally nearly planar regions. Second, we construct a surface triangulation from the output of the clustering step. The output of the clustering step is a set of 2-manifold tiles, which locally approximate the underlying, unknown surface. We construct the triangulation of the entire surface by triangulating the individual tiles and triangulating the ''gaps'' between the tiles. Since we apply point clustering in a hierarchical fashion we can generate model hierarchies by triangulating various levels resulting from the hierarchical clustering step.

In many applications one is concerned with the approximation of functions from a finite set of given data sites with associated functions values. We describe a construction of a hierarchy of triangulations which approximate the given data at varying levels of detail. Intermediate triangulations can be associated with a particular level of the hieracrchy by considering their approximation errors. This paper presents a new data-dependent triangulation scheme for multi-valued scattered data in the plane. We perform piecewise linear approximation based on data-dependent triangulations. Our scheme preserves edges (discontinuities) that might exist in a given data set by placing vertices close to edges. We start with a coarse, data-dependent triangulation of the convex hull of the given data sites and subdivide triangles until the error of the piecewise linear approximation implied by a triangulation is smaller than some tolerance.

We review the cell projection method of volume rendering, discussing back-to-front cell sorting, and approximations involved in hardware color computation and interpolation. We describe how the method accommodates cells with non-planar faces using view dependent subdivision into tetrahedra.

The topology of vector fields offers a well known way to show a ''condensed'' view of the stream line behavior or a vector field. The global structure of a field can be shown without time-consuming user interaction. With regard to large data visualization, one encounters a major drawback: the necessity to analyze a whole data set, even when interested in only a small region. We show that one can localize the topology concept by including the boundary in the topology analysis. The idea is demonstrated for a turbulent swirling jet simulation example. Our concept works for all planar, piecewise analytic vector fields on bounded domains.

In many applications one is concerned with techniques for visualizing data sets with real-time interaction. One technique for providing real-time performance is through the use of multiresolution techniques. These techniques provide multiple representations of a data set at different levels of detail. The idea is to select a level of detail that can be rendered within the user's time constraints. We discuss a mechanism which renders finer-detailed representations where the data set has a high frequency, and coarser representations where the data set has lower frequency. We present a new technique for storing curvilinear data sets within a quadtree representation and discuss two rendering schemes: an anti-aliasing scheme and a scheme for maintaining a specified frame rate.

Betti numbers can be used as a means for feature detections to aid in the exploration of complex large-scale data sets. We present a fast algorithm for the calculation of Betti numbers for triangulated isosurfaces, along with examples of their use. Once any isosurface is extracted from a data set, calculating Betti numbers only requires time and space proportional to the isosurfaces, not the data set. Because the overhead of obtaining Betti numbers is small, our algorithm can be used with large data.

Hierarchical, texture-based rendering is a key technology for exploring largescale data sets. We describe a framework for an interactive rendering system based on a client/server model. The system supports various output media from immersive 3-D environments to desktop based rendering systems. It uses web-based transport mechanisms to transfer the data between the server and the client application. This allows us to access and explore large-scale data sets from remote locations over the Internet. Hierarchical space-subdivision,wavelet compression, and progressive data transmission are used to visualize the data on the client side.